Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT

  • Authors:

  • Jens Gramm;Edward A. Hirsch;Rolf Niedermeier;Peter Rossmanith

  • Affiliations:

  • Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Sand 13, 72076 Tübingen, Germany;Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, 191011 St. Petersburg, Russia;Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Sand 13, 72076 Tübingen, Germany;Institut für Informatik, Technische Universität München, Arcisstr. 21, 80290 München, Germany

  • Venue:
  • Discrete Applied Mathematics - The renesse issue on satisfiability
  • Year:
  • 2003

Quantified Score

Hi-index 0.00

Visualization

Abstract

The maximum 2-satisfiability problem (MAX-2-SAT) is: given a Boolean formula in 2-CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-2-SAT is MAX-SNP-complete. Recently, this problem received much attention in the contexts of (polynomial-time) approximation algorithms and (exponential-time) exact algorithms. In this paper, we present an exact algorithm solving MAX-2-SAT in time poly(L) ċ 2K/5, where K is the number of clauses and L is their total length. In fact, the running time is only poly(L) ċ 2K2/5, where K2 is the number of clauses containing two literals. This bound implies the bound poly(L) ċ 2L/10. Our results significantly improve previous bounds: poly(L) ċ 2K/2.88 (J. Algorithms 36 (2000) 62-88) and poly(L) ċ 2K/3.44 (implicit in Bansal and Raman (Proceedings of the 10th Annual Conference on Algorithms and Computation, ISAAC'99, Lecture Notes in Computer Science, VoL 1741, Springer, Berlin, 1999, pp. 247-258.))As an application, we derive upper bounds for the (MAX-SNP-complete) maximum cut problem (MAX-CUT), showing that it can be solved in time poly(M) ċ 2M/3, where M is the number of edges in the graph. This is of special interest for graphs with low vertex degree.